Although this book is intended, as I indicated at the beginning, to be
a textbook for a science, harmonics is one of those sciences whose
mentality goes back to the earliest times of the history of human
thought, and whose effective constitution is identical with so-called
“Pythagoreanism.” Pythagoras lived in the 6th
century b.c.; thus harmonics, as a characteristic method of study based
on the primal phenomenon of tone-number, has 2,500 years to look back
on. Since all the specific harmonic works of Pythagoreanism are lost,
preserved merely in their rudiments, or have only survived in their
effects upon other domains (mathematics, astronomy, architecture,
grammar, etc.)-this applies to Kepler, A. von Thimus, and V.
Goldschmidt-these observations cover the entire “history of harmonics”
as an autonomous science; and the author's indication of this Textbook
as the first establishment of harmonics as a science, to his own great
regret, cannot be disputed.

Yet the above references show that harmonics, as a characteristic way
of thinking, must have existed in some form since the earliest times,
that it existed at the time of the Pythagoreans, and that its effects
and after-effects, albeit no longer as an autonomous science, have
continually taken shape as a basis for other investigations, up to
those of A. von Thimus and V. Goldschmidt.

The reader interested in history and philology may therefore desire to
see further details, whereupon the author remarks, as he has said all
along, that since he is neither historian nor philologist, he can only
share the information that, so to speak, fell into his hands during the
course of nearly 30 years of harmonic studies. May the following
material inspire a professional to write a real “history of harmonics”!

§55.2. Archaic and Non-Classical Harmonics

The oldest relics of the akróatic mentality are found in almost all
mythologies and fundamental religious principles, as well as in
cosmogonies. Here we must turn our attention to a typical unification
of tone (song, word, speech, hymnal utterance), number (number
mysticism and symbolism), as well as corresponding image-concepts,
especially of an astral nature, which, as we saw in §54, often unveil
themselves in a surprising manner through harmonic analysis. In the
earliest times, “astral mythology” was in a certain sense modern and of
current interest. If one adds to this concept that of the “harmony of
the spheres,” which was widespread throughout Greece, then one has two
terms for very large domains, in which harmonic approaches can offer
things of great promise in analytical and synthetic terms. Also
important are all instances of harmonic numbers, which pervade almost
all ancient religions and cosmological relics, such as the ancient
Indian Rigveda, the Egyptian and Babylonian concept of God, the Zend-Avesta, and above all the wisdom of China.

Here the question emerges of whether, and to what extent, people in
those ancient times knew the specifically harmonic technique: the
monochord. One can unhesitatingly posit it in the most ancient times as
an experimental instrument, due to its simplicity, despite the lack of
concrete proof (which might yet be found in the examination of
archaeological relics, if people only looked for it). If, like the
author, one takes the view that all harmonic forms are psychical
prototypes, which the monochord best “retrieves” from the depths of our
subconscious into waking consciousness, then the search for this
harmonic technique in ancient times would indeed be highly interesting
and exceptionally important for the history of harmonics-but irrelevant
for the factual existence of akróasis in those times. Even if it is
accepted that tone-number investigations were employed here and there
in secret schools (Pythagoras may well have brought his ideas from
Egypt), we must nevertheless take these ancient akróatic relics as
forms that simply correspond to very specific harmonic-psychic
structures and, as long as humanity exists and has existed, continually
strive toward ektypic realization.

However, it appears that people in those early times were already
performing concrete tone-number investigations. Naturally, one must
assume at least a perception for primitive pure-tone ratios as present a priori in humanity. In May 1936, it was demonstrated (Basel National-Zeitung
May 29, 1936) that flutes found at the excavation site at the
mammoth-hunting station of Unter-Wisternitz could play minor and major
thirds, as well as fourths, with a surprisingly clear tone. An ancient
Babylonian baked clay pipe found at Birs Nimrud has the tuning of a
major triad (R. Batka: Allgemeine Geschichte der Musik,
I, p. 39); and from Scandinavia we know of the “Luren,” which were
mostly discovered in groups of two or three, thus showing that
intervals or chords were blown on them. All this would be impossible if
these early people had not already had a developed ability to perceive
tones. With the further progress of culture, and the emergence of
stringed instruments (cf. the Assyrian harps mentioned later),
effective tone-number investigations are almost self-evident. It
appears to me beyond doubt that this was the case in ancient China. In
our analysis of the I-Ching diagram in §50.8, we already mentioned the
works of some French Jesuits, and referred to Windischmann's Geschichte der Philosophie,
in the first volume of which the ancient Chinese doctrine of music and
numbers is discussed, after which there is no question that the Chinese
had a precise familiarity with typical harmonic investigations. To this
is added the enormous significance of the concept of “music” as the
psychic and moral norm. A practical summary of the relevant passages
from Chinese philosophy is given in R. Wilhelm's Chinesische Musik (China-Institut, Frankfurt am Main, 1927) as well as the article by Heinz Trefzger, “Die Musik in China,” in the journal Sinica (year XI 1936, parts 5-6).

In the years following the First World War, I lived in the same
apartment building as an official of Chinese nationality from Eastern
Turkistan, named Burham-Bey. As we got to know each other somewhat
better, one day I showed him and his Chinese secretary my monochord.
Burham-Bey was reminded immediately of an old Chinese man living “near”
Urumqi, who occupied himself with such things. When I asked whether and
how I could get in touch with the gentleman, Burham-Bey protested
almost fearfully: people in that area would receive and read only
letters in Chinese, and not even these if they came from Europe.
Getting a reply was absolutely unthinkable. Furthermore, the man in
question lived in a “fairly” rural place, about 700 km (!) from Urumqi.
This information serves only as proof that number-harmonic
investigation is still alive in modern China (see a further notice in
my Klang der Welt, p. 139). The fascinating book The Music of Hindostan by A.H. Fox Strangways (Oxford, 1914) shows that ancient “Pythagorean” traditions are also still alive in India.

Count Herman Kayserling writes in an extremely beautiful and lively manner in his Reisetagebuch eines Philosophen (6th
ed., 1922, vol. I, p. 398 ff) of a visit to the Tagores in Calcutta,
where he was truly metaphysically “impressed” one evening by authentic
Indian music. But what should interest us especially in Indian music,
from the harmonic viewpoint, is its sensitivity to extremely
differentiated tone-steps. The factual assumptions for the analysis of
these steps are given in §39 (Scales). The theoretician Pavana
identifies six main scale-types and 30 secondary types, and justifies
his system by the fact “that Krishna let five Ragas come out of his
head, his wife Parbuti gave a sixth, whereupon Brahma saw himself
entitled to create 30 more secondary tone-types.” (R. Batka: Allgemeine Geschichte der Musik,
Stuttgart, n.d., I, p. 41). These ragas were ancient melody schemata
which corresponded exactly to times of the year and day, and were
varied creatively by the individual singers and instrumentalists. “Best
known is the Ragavibhoda
[i.e. scale textbook], written by Somanatha in 1609. The description of
the tones in secular music is mostly given with syllables or syllable
signs, and in sacred music mostly with numbers” (Batka, ibid.). If one
then considers that the presumably oldest literary relic, the Rigveda,
is actually a collection of songs, with nothing but the lyrics
remaining for us, whose melodies had specific names with symbolic
meaning and were guarded in strictest secrecy, then we can assume for
sure that number-harmonic investigations were also among these secret
practices.

In Babylonian-Assyrian cultural circles (Neefe: “Die Tonkunst der Babylonier und Assyreer” in Monatshefte für Musikwissenschaft XII, 1890) the harp appears as a symbol and sacred instrument. Friedrich Delitzsch, in his work Babel und Bibel
(1902-1905), showed an Assyrian relief-carving in which the court
“orchestra” can be seen in procession with six great harps. Now, since
one of the three Babylonian classes of priests was that of the singers,
and the Assyrians, culturally dependent upon Babylon, made no exception
in this, and since the technical production of a harp assumes knowledge
of and familiarity with string-length ratios and the corresponding
number-harmonic laws, it is more than likely that the priestly castes
at least had knowledge of primitive harmonic norms. To this is
connected the sexagesimal number system invented in Babylonian-Assyrian
cultures-which corresponds to the harmonic prototype of the
“senarius”-and in fact all of Babylonian numeric and astral symbolism.
“Babylonian number symbolism itself is established beyond all doubt.
Reveries about the value of numbers have a significant place among the
religious philosophical concepts of the Chaldaeans. Each god was
designated by one of the whole numbers between 1 and 60, which
corresponded to his rank in the heavenly hierarchy. A tablet from the
library of Nineveh has preserved for us the list of the most important
gods together with their secret numbers. It appears, in fact, that in
contrast with this scale of whole numbers, which were ascribed to the
gods, another scale of fractions [!] existed, which was connected with
the spirits and corresponded in the same way to their current rank” (M.
Cantor: Vorlesungen über Geschichte der Mathematik, 4th
ed., 1922, vol. I, p. 43). Since the study of these ancient West Asian
cultural circles is in full bloom today, we can expect further
important harmonic discoveries.

Plato, in the second book of his Laws
(Phaidon ed. II, p. 553), tells us that the Egyptians, regarding the
arts, “fixed and consecrated [the forms and strains of virtue] ... And
this practice of theirs suggests the reflection that legislation about
music is not an impossible thing. But the particular enactments must be
the work of God or of some God-inspired man, as in Egypt their ancient
chants are said to be the composition of the goddess Isis.” Besides
harps in all sizes, early flutes and trumpets already appear here.
Diodorus Siculus (I, 16) writes regarding Egyptian priests: “Common
speech had its first development from Hermes, and much that was not
previously designated received its naming. From him comes the invention
of writing in letters, and the arrangement of the worship of the gods
and the sacrifice. He was the first to observe the positions of the
stars and the harmonies and the nature of tones. He invented
swordsmanship and taught the rhythmical movement and the training of
the body to appropriate postures. The lyre he made had three strings,
to signify the three seasons. For he assumed three tones, the high,
low, and middle; the high corresponds to summer, the low to winter, and
the middle to spring. He also taught the Greeks about expression in
speech (hermeneia),
hence his name Hermes. Above all, Osiris employed him as
Hierogrammateus, i.e. as composer and preserver of the holy documents;
people discussed everything with him, and conducted themselves in most
things according to his advice.” This interesting passage shows that
“Hermes” (who appears under various names such as Anubis, Thoth, etc.)
symbolizes “the embodied spiritual life, and with it self-awareness,
thought, teaching, and writing, the genius of the highest science and
wisdom” (Creutzer: Symbolik und Mythologie, 2nd
ed., 1819, vol. 1, p. 363), in contrast and completion to the “natural”
Osiris; and that the ancient form of “akróasis” is realized in the very
symbol of his name. From this alone, setting aside the explicit
references of the ancients to the origin of Pythagoras's wisdom in
ancient Egypt, one can be certain that a definite and specific harmonic
technique (monochord investigations and accompanying diagrams) was
pursued, but was kept strictly secret, and imparted only to initiates.
Thus we can learn nothing from inscriptions, and can only support our
theories upon indirect hints such as the above.

Herodotus, the “father of history,” whose reliability (apart from a few
obvious fables) is increasingly acknowledged, writes in many passages
in the account of his travels of Egypt that he indeed knew more about
some point or other, yet was not allowed to tell it. Reading this, one
has not only the impression of an unquestionable truth, but also of how
the “prohibition” of imparting certain things worked in a constraining
way even on a foreigner. A. von Thimus, at many point in his works,
also gave proofs from classical writers that number-harmonic theorems,
in particular, were deliberately kept secret (see my essay on
Pythagoras in Abhandlungen
for my own views on this secrecy). Compared to this esoteric harmonic
knowledge that was undoubtedly present, the exoteric side of practical
musical education plays a secondary role. Sources include: Ambros: Geschichte der Musik; Lauth: “Über altägyptische Musik” in Sitzungen der bayrischen Akademie 1873; and the older work of Jomard: “Mémoire sur la musique de l'antique Egypte” in Description de l'Egypte,
book III, vol. I, p. 357 ff.). The idea of Memnon, the column of Memnon
that resounds at sunrise, has an entirely harmonic background;
likewise, above all, the ideas dating from this time regarding the
relationship of tone and light (Plutarch, Symposiaca, VIII, 3)-on this see Creutzer's Symbolik und Mythologie, vol. 1, §18.

In Semitic-Arabic cultures, there is firstly the great significance of
music among the Jews (e.g. the Psalms). A. von Thimus devotes large
parts of his Harmonikale Symbolik to number-harmonic analyses of the ancient Hebraic-Kabbalistic book Jezirah;
and the Bible is full of characteristic harmonic number ratios. For
example, the dimensions of Noah's Ark (300 ells long, 50 wide, 30 high)
correspond to the octave-reduced proportion 150 : 50 : 30, i.e. 15 : 5
: 3, which in string lengths forms the major triad (where 1 = c) 15 des 5 as 3 f.
The brilliant Johannes Jacob Balmer, famous for the “Balmer formula” of
the hydrogen spectrum that is fundamental for the modern study of
atoms, was in fact, as I have already mentioned in Klang pp. 84-85, a deeply religious number-harmonist, whose dissertation, Des Propheten Ezechiel Gesicht vom Tempel,
summarized the construction of Solomon's temple according to the
Biblical specifications in word and writing, whose numbers and
proportions have an entirely harmonic character. I was once permitted
to attend an orthodox Jewish funeral. At the lonely snow-covered
cemetery, the main singer, without accompaniment, sang funeral songs of
ancient tradition and completely archaic structure, which made a
profound impression on me that still resounds within me today, due to
their deep psychical content. (Perhaps I heard then a few of those old
Jewish melodies made known by Z. Idelsohn, which bear such an
astonishing similarity to the old Russian church songs that originate
in pre-Christian Greek songs; see Peter Panhoff: “Die Altslavische
Volks- und Kirchenmusik” in Handbuch der Musikwissenschaften, Potsdam 1930, pp. 13-14.)

This kind of spiritual musical expression is possible only when it is
concordant in all respects with the entire spiritual disposition of the
people in question. In Jewry, alone among modern European nations, the
living perception of akróasis of the word has been preserved; compare with this Ben Joseph's important work: “Die Struktur der jüdischen Religionsphilosophie” in Jüdisches Jahrbuch für die Schweiz,
1919-1920, p. 88 ff. As for the Arabic cultural sphere, the history of
harmonics here must first direct its attention to music theory (among
others, see Rosegarten: “Die moslemistischen Schriftsteller über die
Theorie der Musik” in Zeitschrift für Kunde des Morgenlandes,
V). Here the octave is divided into 17 tones, whereby the sharps and
flats are differentiated and the diatonic scale is played with either
sharps or flats alone. This, as well as the general sensitivity of the
Arabs, like the Indians, to finer tone-differentiations, is still
expressed in their modern music-this is shown by any of the muezzin
chants sung from the minarets of the mosques, broadcast often enough on
the radio. Surely a harmonic analysis could clarify this theoretically,
and it would be interesting to pursue the relationship of this
filigree-like music to the geometric “arabesques”; here, too, the
group-theoretical forms of the “P” provide the possible foundations.
Mohammed's religious ecstasies are said to have been accompanied by
ideas of tones, and if he himself only permitted serious, sacred music,
the enormous musical and dance industry of the later caliphs,
especially in Baghdad, shows a strong preponderance of exoteric music.
But a corresponding harmonic study of Arabic philosophy, mathematics,
and astronomy promises, in my opinion, to be even more fruitful.
Regarding the first, I already referred in §50.7 to the encyclopedia of
the “Brethren of Purity,” which preserves a wealth of ancient harmonic
learning. All the ideas of this sect appear to have been influenced in
a Neopythagorean manner, yet somehow developed further independently. A
discussion of the mathematical content of this encyclopedia-admittedly
very rudimentary-appears in Cantor's Vorlesungen über Geschichte der Mathematik, 4th ed., vol. 1, p. 738 ff.-there is also a further bibliography here. Heinrich Suter's essay, Die Mathematiker und Astronomen der Araber und ihre Werke
(Leipzig 1900), names various Arabic authors (no. 63, 116, 198, 303)
who wrote about “music.” The encyclopedias especially (such as that of
Ibn el Khatib, no. 328 of Suter's collection, of which parts still
exist), and the writings of the mystically oriented thinkers, should
one day be studied harmonically, as should the many tracts on
proportions, in which Pythagorean legacies (cf. Nicomachus-Iamblichus)
may have been preserved that are no longer extant in the Classical
legacy. It is known, indeed, that after the destruction of
Constantinople by the Turks, not only many Greek scholars fled,
together with their books, to Arabic or Muslim turf, but also that
shortly afterwards, many Arabic scholars traveled far to the west, and
collected whatever they could find in libraries.

§55.3. Classical Harmonics

After these brief remarks and historical notes regarding a harmonics
that was on the one hand doubtless present but no longer concretely
handed down, on the other hand pervasive of the most varied domains and
cultural circles in its prototypical forms, we now come to
Pythagoreanism as the first emergence in history of harmonics as a
science and philosophy par excellence.
Admittedly, this claim must immediately be dampened by the restriction
that here, we no longer have any of the principal works. However, we do
have concrete fragments (especially Philolaus) and traditions that
situate harmonics beyond question as a science dealing with tone and
number. Here I must refer the reader to my essay on Pythagoras in Abhandlungen,
where the Pythagorean “complex” is considered, admittedly very briefly
and incompletely, but I believe correctly in the essential outline. And
for the first time in the history of research, it is considered from
the viewpoint from which it must be considered: not merely from the
aspect of number, but from that of tone-number. The classical
testimonies that the Pythagoreans used tone-number investigations as
the basis of their studies are so numerous and trustworthy that one can
only marvel at how almost all of modern philology regards the auditory
element, when it does not neglect it completely, as a very unwelcome
and complicated element, whose whimsical nature one must somehow make
allowances for; there is a desire not to place the central Pythagorean
concept of “harmony” on a level with such “lowly” things as monochord
experiments, etc., and it receives a fantastic inflation which no
longer has anything to do with the concrete and flexible sense that
this concept had for the ancients. The otherwise so meritorious Boeckh,
for example, published an essay, “Über die Bildung der Weltseele im
Timäus” (in Daub and Creutzer's Studien, 1806, vol. 3), in which he discusses the famous-and infamous-“Timaeus
scale,” obviously stemming from Pythagorean sources, and very
meticulously seeks to elicit the tone-value of the relevant numbers,
but obviously has not thought of the most important thing: to test
these tone-numbers on the monochord and to hear
them. If one does that, then connections and explanations appear
immediately, as well as corrections, at which purely
logical-intellectual observation alone can never arrive.

I once excitedly brought my monochord to a similarly worthy university
professor, at his wish; he was positively stuffed with knowledge of
sources, references, books, etc. He was working on Pythagoras's
“harmony concept,” and sought to understand the fragments of Philolaus
purely abstractly, on the basis of modern “idealistic” thought. In the
matter of sounding tones and numbers, I explained to him fragments 5
and 6 of Diels, and referred to my work on Pythagoras, in which not
only these, but a series of other important fragments, especially those
on the “apeiron” and “perisson,”
etc., had found obvious clarification. But it was evident that the
learned man was so unmusical that he could not judge the purity of an
octave at all, let alone the fifth, third, or “scale” and their
psychical forms-and he did not want to know anything about my essay on
Pythagoras. When he simply asked me where
I got my insights from, and hoped that I would produce a long list of
literary names and references, I simply indicated the monochord and my
head, as well as the spirit of Pythagoreanism, based on which I
believed myself to have worked and thought. He smiled pityingly: “Well,
well!” I tell this story only as characteristic of the modern
situation, which every harmonist will find himself confronting in
regards to science! The clogging up of what has been handed down from
ancient times, especially in historical things, is so great that no
number of Galilean telescopes can help, and we must wait for new
spiritual powers outside the “science” that will take up and continue
the new ideas with enthusiasm, and without worrying about success or
failure. The only book I have found after the completion of this
Textbook that seeks to grasp Pythagoreanism, and above all the
“Pre-Socratics” based on the nature of these thinkers themselves, is K.
Joel's excellent work: Der Ursprung der Philosophie aus dem Geiste der Mystik (Jena, Diederichs, 1906), which is recommended especially to the reader.

For a new foundation of the spiritual history of Pythagoreanism, beside the standard works-Harmonikale Symbolik
by Baron A. von Thimus (2 vols., Cologne 1868-1876) and the above
mentioned book by K. Joel-I know of only two useful books in the German
language: Erich Frank's Plato und die sogenannten Pythagoreer (Halle 1923) and Julius Stenzel: Zahl und Gestalt bei Platon und Aristoteles
(Leipzig-Berlin 1924). In neither work is “harmonics” considered as an
autonomous concept, and thus Thimus's work was also unknown to them.
Nonetheless, Frank is one of the few scholars who views the enormous
significance of music for the ancient Greeks in the correct light, and
the information, references, etc. that he gives are very useful as an
initial basis; his idée fixe that
Pythagoras never actually existed is unimportant for our purposes.
Stenzel begins more with the “dieresis of ideas,” i.e. actually with a
demonstration of ancient thought with-as we may express it
harmonically-the “law of harmonic quantization,” which in fact
manifests from the monochord in the “P” system. As a basis for the
ancient sources, the literature about it, etc., the following things
are indispensable: (1) the first volume of Überweg's Geschichte der Philosophie, and (2) Diels' Fragmente der Vorsokratiker.
Especially in the latter work, the reader learned in Greek will find,
in the information, quotes, etc., surrounding the fragments, a wealth
of harmonic treasure, which can now be summarized, since harmonics has
been newly constituted, from unified viewpoints. Admittedly, as I
learned from Olof Gigon: Der Ursprung der griechischen Philosophie (Basel, Benno Schwabe, 1945, p. 11), Diels' Fragmente der Vorsokratiker
offers only a restricted choice of texts; a future study of Pythagoras
should first assemble all the material that has been handed down to us.

If this-the rectification of true Pythagoreanism-ever happens, then we
will also see that his influence reached much farther than people have
yet dared to believe. Not only music and astronomy must be included in
this: not only arithmetic, geometry, grammar, rhythm, and above all
mythology, Neopythagoreanism, Gnosticism, architecture, etc. receive
new illuminations from it; but all these domains are finally truly
understandable from an inner, central synthetic viewpoint, namely that
of akróasis, above all in the deeper sense as the collective cultural
and spiritual attitude of an epoch. Thus it will also be shown that we
must give up the comfortable humdrum of simply denying anything for
which there is no explanation, all that we do not understand or that
does not fit into the common philosophical-historical judgment. A model
example of this is Euclid's essay “On the Division of the Canon” (for
the ancients, “canon” was synonymous with “monochord”), long seen as a
“falsification.” How could such a great man devote his time to such an
outmoded thing as the division of the monochord! People do not
understand, or do not want to understand, that at that time the
monochord was a teaching tool precisely for the knowledge of the study
of proportion that was so important for the ancients, completely aside
from its value as the essential
Pythagorean experimental instrument. We know from “legend” (!) that
Pythagoras, shortly before his death, asked one of his favorite
students to strike the monochord, “thus indicating,” as Aristides
Quintilianus writes in his De Musica,
“that the highest and final things that music treats can be grasped not
so well by means of the tones heard through sensory perception, as by
way of intellectual observation of the numbers” (from Thimus I, 128).
Erich Frank (op. cit., p. 182) thus writes with complete justification
of Euclid's work: “This Canon Division can thus be understood as a
counterpart to Euclid's Elements: while the latter provides the
mathematical knowledge necessary for the Platonic construction of the
world's body, the laws at the basis of the construction of the world's soul are developed in the canon.” In the same breath (pp. 183-184), however, Frank explains the “scale of the [Platonic] Timaeus”
as a “theory born from the start as a dead letter,” indeed as “insane
number speculation” and a “musically quite impossible scale” (p. 17).
Thus the same thing happened to him that happened to all commentators
before him (Tannery etc.): he did not understand this “scale” at all.
A. von Thimus (Harmonikale Symbolik I, 156 ff. and II, 281 ff.) gave the only adequate interpretation of the Timaeus
scale thus far, from his all-embracing knowledge of ancient
number-harmonics and Pythagoreanism (see the illustration of this scale
in §13a and §39.2a of this book)-of course, without eliciting the
slightest result for those who study it. The reason is, as with all
harmonic discoveries and references, always the same: people expect
recipes, but they should realize that the necessary prerequisite of all
“harmonicalia” is not only a precise familiarity with Greek language,
arithmetic, and music, but also an equally precise knowledge of the
harmonic technique, which must be learned and studied just like the
grammar of any language. The otherwise so thorough and philologically
precise E. Frank writes, for example, the following (op. cit., p. 154):
“The basis of our modern music is the diatonic [!] octave, where
between two tones there is always [!] a whole-tone interval [!] and
hence [!] this type of tone has been thus named by the Greeks [!].”
Armed with such a “knowledge” of elementary matters, he then writes for
pages about Greek music theory, and projects this nonsense in his head
onto the “insane and musically quite impossible scale of Timaeus”!
Now, since a musical historian typically understands nothing of
mathematics, the philologist nothing of music or mathematics, and the
“classical” historian of philosophy nothing of any of the three
subjects, and none of the three know or want to know anything about the
technical-harmonic fundamentals of harmonics (monochord laws), we still
have the persisting calamity of a misunderstanding of precisely such
problems as the Timaeus scale, ancient Greek enharmonics, and so forth-even long after their solution has been found (by Thimus).

Observed, or rather newly observed, from the point of view of akróasis,
not only Pythagoreanism, the pre-Socratics, Plato, and Aristotle, but
also the entire “succession” of the ancients must be subjected to a
corresponding revision.

As the central workbook for the ancients and their precursors, for a future history of harmonics, A. von Thimus's Harmonikale Symbolik
is above all to be valued and used. In §25.1 I have tried to
characterize the value and majesty of this work in general terms.
Thimus's harmonic “toolbox” is admittedly limited in
intellectual-discursive terms and ignores many geometric-visual
theorems that are important and indeed indispensable for an
illumination of many ancient symbols. The reader will easily be able to
form his own judgment after a careful comparison of Harmonikale Symbolik
with this Textbook. However, Thimus presents an almost endless array of
historical material which should now be worked through anew from the
point of view of harmonics as an autonomous science and way of
thinking.

It will be best, in what follows, to discuss the effects of harmonics
in the individual domains-always naturally under the restriction of the
author's less than adequate knowledge in historical matters, and only
considered as catchword-like pointers for a future “history of
harmonics.”

§55.4. Music Theory

Firstly, ancient music theory and its succession. Its kernel is the
ancient Greek “enharmonics.” This was first discovered in all its
richness, unveiled, and precisely examined, by A. von Thimus, and
anyone who has closely studied the relevant parts of Harmonikale Symbolik,
and after this hard-earned knowledge has taken up some book on Greek
music theory, will agree with me that we must start again from the
square one; that all other authors, no matter how worthy their work may
be (Westphal, Ambros, Riemann, Abert, etc.) are either completely
ignorant or work with false assumptions about this central concept of
enharmonics (which makes chromatics, diatonics, rhythm, etc.
understandable for the first time). Thimus devoted his life's work to
this, and any future revision of this domain that ignores what he has
offered is, like everything written since his work, condemned a priori to
obsolescence. If these ancient fundamentals explained by Thimus are
taken up anew, then new light falls upon all of music theory from the
Middle Ages to modern times. It is more than merely accidentally or
“coincidentally” significant that the ancient Pythagorean experimental
instrument, the monochord (see Wantzloeben: Das Monochord als Instrument und als System,
Halle 1911), has been used right up to modern times as a scientific and
practical teaching instrument (see §1c and §1d). It is the symbol for a
pervasive, living Pythagorean legacy, and I am convinced that from the
numerous writings about the monochord (see Wantzloeben's bibliography)
and the ancient harmonics that still sound in it, entirely new
viewpoints will arise not only for the history of music theory, but
also for the related practical domains such as notation, church music,
the study of instruments, etc.

§55.5. Mathematics

Next, the fundamentals of mathematics. Cantor writes (Vorlesungen über Geschichte der Mathematik, 4th
ed., vol. 1, p. 153): “We believe we are justified in the association
of the name of Pythagoras with the musical study of numbers; whether or
not the monochord originates from him, we believe that he occupied
himself mainly with the arithmetical subdivision of geometry.” Diogenes
Laertius writes (VIII, 12): “µ???stades????se?t??p??a???a?pe??t??a???µ?t???????d??a??t?~? (sc. ?e?µ?t??a?) t??de?????at???e?µ?a????d?~?e???e?~?.”
Precisely this, the derivation of arithmetical and geometrical forms
from the monochord, was discovered by Thimus (I, X, 65, 118 ff., 126,
218, 249 ff., 350, and II, 43 note) in the first book of Nicomachus's Arithmetic
and the accompanying commentary by Iamblichus-two texts that are not
sufficiently respected, either by musical science or by mathematics.
From them, Thimus developed the scheme of our partial-tone coordinates,
and it is beyond doubt that in this scheme we have the enigmatic
“Pythagorean table” and, mutatis mutandis, the later “abacus.”

In Boethius's geometry (ed. Friedlein, pay., 395-396) we find the following interesting passage:

“Men
of ancient insight, who belonged to the Pythagorean school and occupied
themselves as searchers into Platonic wisdom with remarkable
speculations, have placed the peak of all philosophy in the
peculiarities of numbers. In fact, who will understand the measures of
musical harmony if he believes that they are not connected with
numbers? ... The Pythagoreans, so as not to get lost in errors in
multiplications, divisions, and measurements (as they were full of
brainwaves and refinement in all things), availed themselves of a
certain drawn figure, which they named, in honor of their teacher, the
Pythagorean table (mensa Pythagorea),
because the first teaching of things thus represented had come from
this master. Later peoples knew the figure as Abacus. With this they
intended to bring profound ideas more easily to general knowledge, as
when one sees it before one's eyes, and gave the figure the following
remarkable form.”

(from
Cantor, op. cit., pp. 583-584.) However, this figure, which then
follows in the manuscripts, is simply an ordinary multiplication table,
as Thimus mentioned (vol. I, table II, Fig. 1 and text p. 144 ff.), and
has not the least to do with “profundity.” It is to be assumed that by
the mensa Pythagorea,
the ancient Pythagorean “Lambdoma” was meant, and therefore the
partial-tone coordinates that even Boethius did not know correctly but
only from legend, and which indeed, besides their musical norms, also
contain very important arithmetical and geometrical laws, as we have
shown in several passages in this book: laws that go far beyond the
multiplication table and deeply into number-theoretical speculations
(for which such a mathematically gifted people as the ancient Greeks
had a great understanding). The foundation on a harmonic basis of
arithmetical, geometric, and certain number-theoretical elements
(concept of the infinite, the irrational, as well as the entire study
of proportion that was so important to the ancients) is beyond question
for anyone who arrives by necessity via monochord experiments upon the
diagrammatic notation of the “P” and insight into its number-harmonic
construction, and its obvious arithmetical, geometric, and
number-speculative configurations. The famous “discovery of Pythagoras”
of the dependence of tone-perceptions (quality) on string lengths
(quantity) in the sense of a precisely determinable numerical
relationship almost pales beside this, being reduced to an admittedly
important “special case” of Pythagoreanism, when we are forced to see,
in this discovery, the birth of our precise scientific methods. But as
I always emphasized regarding this discovery (which was certainly
ancient knowledge in all high oriental cultures!), its “flip side” must
be seen as at least as important for ancient culture: that in this
discovery, the quantitative (string length, or simply matter) can be
qualitatively-psychically evaluated (the audible tone ratios). And it
was precisely this double aspect of the tertium comparationis
of the “number”-that it reaches into matter on one side, into our
psyche on the other side, namely into our psyche not only as
intellectual-discursive logical form, but as an entire gestalt of our
perception, feeling, our soul (interval, chord, scale, etc.)-which so
“intoxicated” the ancients in this “discovery of Pythagoras.” This
akróatic background of ancient number-thought has been completely lost
to us today, and for these reasons we also find, in almost all works on
the history of mathematics, a complete lack of understanding, and
connected with it, a lack of interest regarding the harmonic
foundations of ancient mathematics. Here almost everything must be
redone and rebuilt.

Figure 479

The partial-tone coordinates of index 16 (PE16) with their logarithms (base 2), coordinates and tone-values, decimals and angles (frequencies).

§55.6. Language, Grammar, Rhythm

Next, the foundations of language (word), grammar, and rhythm. The
Introduction to this book discussed the akróasis of the word, speech,
and the harmonic background of all spiritual utterance given a priori
with it. This background was expressed in a peculiar way in the earlier
times of high cultures through the singing, or reading in an emphatic
voice, of the cultic hymns and songs, and it has continued in the
rituals of all religions to date. The “music” here is not only an “art”
like painting or architecture, which latter gives worship more of its
outward symbolic consecration. The musical element here is deeply
connected with the metaphysical meaning of the “word” as that form of
communication that mediates the divine by auditory means.

Music and word, seen thus, are finally the same: pronouncements from
and to God. If I now refer briefly to the specifically harmonic
backgrounds of ancient grammar and metrics (the study of syllables,
etc.), it is on the one hand with the belief that here harmonics still
has much to offer, but on the other hand with the assumption that in
philosophical regards (including by the ancients) much effort has been
made of which I am still unaware. As for grammar, I simply refer to
Eberhard Hommel (Untersuchungen zur hebräischen Lautlehre,
part I: “Der Akzent,” Leipzig 1917). I have already quoted a passage
from this in §31a, and I entreat the reader to read it once more. These
“threads” must be followed further by specialists. Important material
related to this can also be found in Franz Dornseiff: Das Alphabet in Mystik und Magie, Leipzig 1922. In Plato's late dialogue, the Philebus, there
are observations on tone, number, sound, and letters, which refer to
the great age (Egypt) of these typical cross-references. A future
historian of harmonics will have an easier time with the connection of
harmonics to rhythm and metrics. I know of two works, (1) Aristides
Quintilianus: Über die Musik (tr. and thorough commentary by Schäfke, Berlin 1937), and (2) the church father Augustine, Musik
(tr. by Perl, Strasburg 1937), which contain extensive rhythmic and
metric observations, together with their relationships and derivations
to and from harmonic number-ratios, which can easily be developed
further, both forward and backward. Moreover, these particular works
have a truly Pythagorean “timbre,” i.e. an akróatic spiritual
disposition pervaded by the universal significance of the musical,
which in many places gives one pause and fills one with awe and
astonishment at such a depth of thought-unfortunately entirely lost
today.

§55.7. Astronomy, Astrology, the Harmony of the Spheres, Astral Symbolism

Next, astronomy, and the astrology allied with it, as well as the
harmony of the spheres and astral symbolism that emerge from both.
Copernicus, in his De revolutionibus orbium coelestium (1543),
explains expressly that he has taken up the heliocentric solar system
of the Pythagoreans. He quotes Plutarch: “The others believe, however,
that the Earth stands still, but the Pythagorean Philolaus says that it
moves around the fire on the slanting circle of the ecliptic in the
same direction as the sun and the moon.” “Starting from here, I
[Copernicus] began to contemplate the movement of the Earth and,
although the viewpoint seemed absurd, I still did it, because I knew
that others before me had already been granted the freedom of
conceiving arbitrary circular movements for the representation of the
heavenly bodies” (from E. Frank: Plato und die sogenannten Pythagoreer,
Halle 1925, p. 37). Frank presumes (p. 38) that the origin of the
heliocentric idea is to be sought among the writers associated with
Archytas. Archytas, like Democritus, wrote a work on harmonics that is
now lost, and both appeal to Pythagoras as the ancestor of their ideas.
A few fragments of Archytas's writings still remain, which O.F. Grupps
published (Über die Fragmente des Archytas,
Berlin 1840) and, following the skeptical custom of his time, declared
that “not a single thing is to be taken as true.” This untenable
viewpoint has been taken ad absurdum by A. Speiser in his translation of the beautiful fragment from the harmonics of Archytas (Klassische Stücke der Mathematik,
Zürich 1925, pp. 9-11). Archytas's principles still resound in the
harmonics of Ptolemy, according to E. Frank (op. cit., p. 166).
Claudius Ptolemy, mainly known for his Syntaxis (the “Almagest”), wrote, among other things, a still surviving Harmonics,
which, paraphrased by Porphyry in another surviving commentary
(Ptolemy's study of harmonics and Porphyry's commentary were published
in two excellent text-critical editions by Ingemar Düring, Göteborg
1932), was and quoted numerous times by Johannes Kepler in his Harmonice Mundi
(see the entries “Ptolemäus” and “Porphyrius” in the index of Caspar's
translation), and mentioned expressly by Kepler as one of his most
important sources and inspirations. Kepler himself originally had the
intention of commenting exhaustively on Ptolemy's Harmonics, and a brief part of this commentary is included in his Opera omnia.
Due to my limited knowledge of the Greek language, I have not yet been
able to study the relevant works of Ptolemy and Porphyry; surely they
contain important material for the history of classical harmonics.

Little known and less investigated are the relationships of harmonic to
astrology and partly also, via astrology, to alchemy. Very early on,
the doctrine of aspects was already compared with intervallic
consonance and dissonance. In the fourth, astrological book of his Harmonice Mundi
(Chapter 5), Kepler, after a most interesting exposition on the
intelligible nature of the harmonies (musical philosophy has let these
valuable ideas escape it until now), discusses the actual doctrine of
aspects, based on Ptolemy, Cardan, and Reinhold, tracing the musical
intervals and astrological aspects to geometrical phenomena.

The astrological “Inventory” of all of alchemy, the association of the
planets with the elements, the calculation of the correct aspects for
alchemical operations, etc., is so well known that this reference alone
suffices. But what historians have overlooked until now with all these
“analogies,” what makes them finally understandable in a concrete
harmonic sense, and what one can only grasp by means of harmonic
thought, is the spontaneity-as I might express it-with which the
interval is executed. Just as this is expressed in the “purity” or
“impurity” of the interval (the smallest variations “disturb,” for
example, the octave, the fifth, etc.), likewise the aspect requires a
precise execution, i.e. a precise calculation. Likewise, the correct
execution of an alchemical transmutation can only be successful at the
“favorable” hour, i.e. in an interval that is consonant between planet
and element. The execution of every chemical bond shows that such ideas
are by no means merely illusions. The so-called “law of multiple
proportions” of modern chemistry is nothing other than the acceptance
of certain weight intervals, which must first be separated with all
possible precision from the masses of individual substances before they
enter into the execution of the new bonding! (As literature I recommend
A. Fankhause: Das wahre Gesicht der Astrologie, Zürich 1932, and C.G. Jung's work, supported by vast knowledge: Psychologie und Alchemie, Zürich 1944.)

I can only refer to the broad domains of the harmony of the spheres and
astral symbolism as historical domains which, with the new
harmonic-analytic method achieved on the basis of this book, can be
dealt with in terms of their actual inner nature. Far into the Middle
Ages, there was an enormous body of literature, which for the most part
still remains buried in the manuscripts of libraries, and has not yet
been edited at all-the recently published catalogs of medicinal
manuscripts are proof enough of this. Regarding the harmony of the
spheres, Jacques Handschin published an excellent essay in the Zeitschrift für Musikwissenschaft
1926-27, “Beitrag zur Sphärenharmonie,” which can be used as a basis
for further studies due to its deeply based erudition. The literature
on astral symbolism is substantially larger-here I will mention only
F.X. Kugler: Sternkunde und Sterndienst in Babel (1907-1924) and R. Eisler's work, already cited many times: Weltenmantel und Himmelszeit (Munich 1910), in which the interested reader will find a wealth of material and further reading.

§55.8. Architecture and Visual Art

Regarding the history of architectural harmonics, the most important
things have been indicated in §29 of this book, so I can summarize
quickly here. The central works here are those of Vitruvius and
Eichhorn. Harmonics in painting and sculpture, as in architecture, has
its background in certain proportions. There are three “primal
proportions,” the “arithmetic,” the “harmonic,” and the “geometric”
proportion, which, as we saw in §28, are all contained in the “P”, and
are thus of harmonic nature. Thus, when one writes of the history of
the harmonics of architecture and the visual arts, one must preface it
with the history of proportions and of proportional technique, which
was of enormous importance in classical cultures. Thimus anticipated
the essentials in his “preamble” (from the commentary of Iamblichus on
Nicomachus's Introduction to Arithmetic) and, in the course of his Harmonikale Symbolik,
he developed the proportional technique found there most precisely and
thoroughly-a very subtle method, not at all simple for us today, since
we have become so “non-visual” compared with the ancients. This
proportional technique should be handled and expounded in its
historical metamorphosis up to the Renaissance as a fundamentally
separate domain of mathematics; one will then not only see that the
study of proportion has directly stimulated architecture and painting
through all periods (until its downfall in modern times)-think only of
the proportional studies of Leonardo da Vinci, Albrecht Dürer, as well
as almost all great architects!-but one will also then be cured of the
one-sidedness already mentioned in §28, such as the “golden section”
etc. (which fanatics have meanwhile built into disproportionate
complexes). The three primal proportions that were placed at the peak
by the ancients-the arithmetic, harmonic, and geometric proportions-are
harmonic by their very nature, and constitute such a wealth of formal
possibilities, that in contrast, those individual proportion types such
as the Golden Section, the p/n
triangles, certain circle divisions, and so forth, appear sterile and
poor. But these three primal proportions are concentrated in the
“harmonic division canon” of the “P”, which, as I have shown in my work
on Villard (Harmonikale Studien,
vol. 1), must have still been known in the Gothic era as a Pythagorean
legacy (see §38.1a and §41.4 of this book). The individual harmonic
analyses of the individual styles must then be built upon this harmonic
division canon; and Villard's problem in particular has convinced me
that this canon, within the Gothic style itself, provides a method of
style-analysis that will yield valuable results.

§55.9. Philosophy and Symbolism

Historical information about harmonic relationships to philosophical
and epistemological phenomena can be found in many chapters of this
book, and regarding symbolism-especially religious and cosmological-the
interested reader will find material sufficient for a start in §54. A
historic view of harmonics in these domains is synonymous with the
history of the great spiritual stages of harmonics. Therefore I will
only mention the key terms: archaic harmonics (especially China),
Pythagoreanism (fragments of Pythagoras, Aristides Quintilianus,
Ptolemy), Plato's late philosophy (Timaeus scale), Augustine (De musica),
various Renaissance philosophers only known to me by name, such as
Marsilio Ficino, Cardan, etc., in whose works harmonicalia can
presumably be found; also Robert Fludd should, despite Kepler's
polemic, at least be examined in a historical-harmonic sense; then of
course, above all, Kepler himself, the Harmonie universelle of Father M. Mersenne, Paris (3 vols., 1644-1647), Athanasius Kircher's Musurgia;
of the more recent, especially Leibniz, Th. Fechner, J.J. Bachofen, and
last and still most important for us, A. von Thimus. From these
writings, details and cross-references will arise of themselves, indeed
entirely new names and works will emerge, which have not yet been
historically placed anywhere, and will perhaps for the first time find
their homes in a “history of harmonics.” For the history of harmonic
symbolism, Thimus is authoritative above all: the introduction to his
work, especially, gives a wealth of names and references that are
indispensable not only for symbolism, but for the historical
development of the number-harmonic way of thinking in general.

§55.10. Conclusion

I have come to the end of this work. Kepler wrote, on July 28, 1619, to Lord Napier, after finishing Harmonice Mundi: “Harmonics is finished, thanks to the grace of the highest harmost
of the universe. In vain the god of war has rumbled, crashed, and
yelled with bombs and trumpets and his whole tarantara. If the fury of
war does not besiege us at home or outside, or tear away the workers
and leave us stranded, everyone can come and see the copies of Harmonics,
and my work on the comets, at the coming Autumn Fair in Frankfurt, all
those whose hearts desire to observe the works of God's hands more
deeply, as I have illuminated them with the light of understanding.”

At the hour that I write this, the heavy thunder of the death-machines
of this world war drones from the direction of Basel-and when I think
of my ancestors who, along with Kepler, left their homeland for reasons
of faith to seek asylum in foreign parts, this “repetition of history”
seems to me, in my asylum country of today, more than merely the proof
of a harmonic theorem. Without daring to make a comparison with the
completion of Kepler's work,
the friendly reader will understand and be lenient to me when I
conclude with Kepler's words and dedicate this book, as a small
building block for the reestablishment of our poor, ravaged Europe, to
all those “whose hearts desire to observe the works of God's hands more
deeply, as I have illuminated them with the light of understanding.”